AbstractWe prove the global $$L^p$$
L
p
-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $$S^{m}_{\rho , \delta }(\mathbb {R}^n)$$
S
ρ
,
δ
m
(
R
n
)
for parameters $$0\le \rho \le 1$$
0
≤
ρ
≤
1
, $$0\le \delta <1$$
0
≤
δ
<
1
. We also consider the regularity of operators with amplitudes in the exotic class $$S^{m}_{0, \delta }(\mathbb {R}^n)$$
S
0
,
δ
m
(
R
n
)
, $$0\le \delta < 1$$
0
≤
δ
<
1
and the forbidden class $$S^{m}_{\rho , 1}(\mathbb {R}^n)$$
S
ρ
,
1
m
(
R
n
)
, $$0\le \rho \le 1.$$
0
≤
ρ
≤
1
.
Furthermore we show that despite the failure of the $$L^2$$
L
2
-boundedness of operators with amplitudes in the forbidden class $$S^{0}_{1, 1}(\mathbb {R}^n)$$
S
1
,
1
0
(
R
n
)
, the operators in question are bounded on Sobolev spaces $$H^s(\mathbb {R}^n)$$
H
s
(
R
n
)
with $$s>0.$$
s
>
0
.
This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.
Boundary value problems for nonlinear partial differential equations in anisotropic Sobolev spaces